@scrabblerIn how many ways can four persons be selected from ten persons sitting around a circular table such that none of the selected person were sitting next to each other previously?A. 10 B. 15 C. 25 D. 100
I make it 25. I actually enumerated :P
Let us call the ten people 1, 2, 3, 4, 5...10
Consider when 1 is selected (here 10 can't be selected):
1, 3, 5, ke saath 789 (3 cases)
1, 3, 6 ke saath 89 (2 cases)
1, 3, 7 (1 case)
1, 4, 6 (2)
1, 4, 7 (1)
1, 5, 7 (1)
Now if 1 not chosen, then starting with 2 (here 10 can be selected)
again we find
2, 4, 6 (3 cases)
2, 4, 7 (2)
2, 4, 8 (1)
2, 5, 7 (2)
2, 5, 8 (1)
2, 6, 8(1)
With 3
3, 5, 7 (2)
3, 5, 8 (1)
3, 6, 8 (1)
With 4
4, 6, 8 (1)
Total 25. I am sure there must be an easier way but not seeing it right now...
@scrabblerIn how many ways can four persons be selected from ten persons sitting around a circular table such that none of the selected person were sitting next to each other previously?A. 10 B. 15 C. 25 D. 100
Select 1 person among 10 -> 10 ways
Select 3 person among remaining 9 such that none are adjacent -> this is now same as selecting 3 person out of 9 in linear arrangement, such that none are adjacent
So this can be done in C(5,3) ways (same as placing 3 people between the 6 where 6 people have 5 gaps between them)
So total ways = 10*C(5,3)
Now each quadruple has been repeated 4 times since we have 4 people that we have picked up.
So total unique cases = 10*C(5,3)/4
One can generalize this solution in that if we are asked to form a polygon of n 'sides' using the vertices of a larger polygon of N sides such that no two vertices are adjacent, then total ways = N/n * C(N-n-1, n-1)
Two players play a game using the interval [0,33] on the x-axis. The first player randomly chooses a square of side length s∈Z+, which has a side that lies entirely on the interval. The second player randomly chooses a circle with radius r∈Z+, which has a diameter that lies entirely on the interval. After repeating choosing random squares and circles in this fashion, the players realize that the probability that the circle and square intersect is 1/2. Let S={(s,r):p probability of intersection is 12}. Determine ∑(s,r)∈S {(s+r).}
Clarification of notation: The set S is the set of all ordered pairs of integers, (s,r), such that the probability that a square of side length sand a circle of radius r will intersect is 1/2.
What is minimum value when three values are added?? Does the minimum value occur at a point where all entities added are equal? What is rule of symmetry in maxima and minima? Please shed some light on these above topics pleaseeeee??
@scrabblerIn how many ways can four persons be selected from ten persons sitting around a circular table such that none of the selected person were sitting next to each other previously?A. 10 B. 15 C. 25 D. 100
open the circle making a straight line
x1 - x2 - x3 - x4 -x5
x1 and x5 can be 0 but x2 x3 x4 can not
x1+x2+x3+x4+x5 = 6 => 7c4 = 35 ways
but we need to exclude cases when both persons at extreme ends selected, because when it makes circle these persons will be adjasent
in this case 2 people adjascent to erxtreme end can not be selected
so we left with 6 option
out of these 2 people are to be selected who are not adjasent
question 2 .. The number of ways in which we can select 5 numbers from the set of numbers(1,2,3,4,5.......25) such that none of the selection include any two consecutive number?
At the end of MAHABHARAT, Pandav's family is saying goodbye to Kaurav's family. Each member of Pandav's family says goodbye to each member of Kaurav's family. To say goodbye, two men shake hands and a man and a woman and two women kiss once on the cheek. An eyewitness to the event counted 21 handshakes and 34 kisses. If there are at least one man and one woman in each family, then how many MEN and WOMEN were there?
At the end of MAHABHARAT, Pandav's family is saying goodbye to Kaurav's family. Each member of Pandav's family says goodbye to each member of Kaurav's family. To say goodbye, two men shake hands and a man and a woman and two women kiss once on the cheek. An eyewitness to the event counted 21 handshakes and 34 kisses. If there are at least one man and one woman in each family, then how many MEN and WOMEN were there?
suppose a men in Pandavas family and b men in Kaurav's family
c women in Pandav's family and d women in Kaurav's family
I am sure there must be an easier way but not seeing it right now...
I am sure there must be an easier way but not seeing it right now...